arXiv subjects
Publications and source records attributed to Chris Parker.
This article presents a 3-local characterisation of the sporadic simple group McL and its automorphism group. The theorem is underpinned by a further identification theorem the proof of which is character theoretic. The main theorem is applied in our investigation of groups with a large 3-subgroup. An additional file containing Magma code has been attached to this submission.
We construct a family of finite special 2-groups which have commuting graph of increasing diameter
The commuting graph of a finite soluble group with trivial centre is investigated. It is shown that the diameter of such a graph is at most 8 or the graph is disconnected. Examples of soluble groups with trivial centre and commuting graph of diameter 8 are provided.
We present an identification theorem for the groups $F_4(2)$ and $Aut(F_4(2))$ based on the structure of the centralizer of an element of order 3.
The main theorem in this article shows that a group of odd order which admits the alternating group of degree 5 with an element of order 5 acting fixed point freely is nilpotent of class at most two. For all odd primes r, other than 5, we give a class two r-group which admits the alternating group of degree 5 in such a way. This theorem corrects an earlier result which asserts that such class two groups do not exist. The result allows us to state a theorem giving precise information about groups in which the centralizer of every element of order 5 has centralizer a 5-group.
We identify the sporadic groups M(23) and F_2 from the approximate structure of the centralizer of an element of order 3.
For a prime $p$, a $p$-subgroup of a finite group $G$ is said to be large if and only if $Q= F^*(N_G(Q))$ and, for all $1 \neq U \le Z(Q)$, $N_G(U) \le N_G(Q)$. In this article we determine those groups $G$ which have a large subgroup and which in addition have a proper subgroup $H$ containing a Sylow $p$-subgroup of $G$ with $F^*(H)$ a group of Lie type in characteristic $p$ and rank at least 2 (excluding $\PSL_3(p^a)$) and $C_H(z)$ soluble for some $z \in Z(S)$. This work is part of a project to determine the groups $G$ which contain a large $p$-subgroup.
We show that the sporadic simple group $\M(22)$, the exceptional group of Lie type ${}^2\E_6(2)$ and their automorphism groups are uniquely determined by the approximate structure of the centralizer of an element of order 3 together with some information about the fusion of this element in the group.
We identify the groups $PSU_6(2)$, $PSU_6(2):2$, $PSU_6(2):3$ and $Aut(PSU_6(2))$ from the structure of the centralizer of an element of order 3.
We generalize a result of Serre's to show that if every vertex of some fixed type of a convex subcomplex of an irreducible spherical building has an opposite, then the subcomplex is completely reducible.
Conway's second largest simple group, $\Co_2$, is characterized by the centralizer of an element of order 3 and certain fusion data.
Suppose that $p$ is a prime, $G$ is a finite group and $H$ is a strongly $p$-embedded subgroup in $G$. We consider the possibility that $F^*(H)$ is a simple group of Lie rank 2 defined in characteristic $p$.
We study finite groups which possess a strongly p-embedded subgroup for some odd prime p. The main results of the paper will be applied in the ongoing project to classify the simple groups of local characteristic p.
We identify the sporadic simple group $\mathrm{M}_{12}$ and the simple group $\mathrm{SL}_3(3)$ from some part of their 3-local structure and give a graph theoretic analogue of the resulting theorem.
The groups which can act semisymmetrically on a cubic graph of twice odd order are determined modulo a normal subgroup which acts semiregularly on the vertices of the graph.